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Comparing The Models SARIMA, ANFIS And ANFIS-DE In Forecasting Monthly Evapotranspiration RatesUnder Heterogeneous Climatic ConditionsPouya Aghelpour
Bu-Ali Sina UniversityVahid Varshavian ( [email protected] )
Bu Ali Sina University Faculty of Agriculture https://orcid.org/0000-0002-9705-3066Zahra Hamedi
University of Birmingham
Research Article
Keywords: Differential Evolution, ANFIS, Stochastic, ARIMA, Time series prediction, Reference CropEvapotranspiration
Posted Date: August 17th, 2021
DOI: https://doi.org/10.21203/rs.3.rs-781601/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License
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Comparing the models SARIMA, ANFIS and ANFIS-DE in 1
forecasting monthly evapotranspiration rates under heterogeneous 2
climatic conditions 3
4
Pouya Aghelpoura, Vahid Varshavianb*, Zahra Hamedic 5
(a) MSc graduated of agricultural meteorology, Department of Water Engineering, Faculty of Agriculture, 6
Bu-Ali Sina University, Hamedan, Iran (Email: [email protected]; ORCID: 7
https://orcid.org/0000-0002-5640-865X) 8
(b*) Assistant Professor of agricultural meteorology, Department of Water Engineering, Faculty of 9
Agriculture, Bu-Ali Sina University, Hamedan, Iran (Email: [email protected]; ORCID: 10
https://orcid.org/0000-0002-9705-3066) 11
(C) MSc graduated of Computer Science, Computer Science Department, University of Birmingham, 12
Birmingham, UK (Email: [email protected]; ORCID: https://orcid.org/0000-0001-5279-13
500X) 14
*Corresponding author name: Vahid Varshavian 15
*Corresponding author Email: [email protected] 16
17
18
19
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Comparing the models SARIMA, ANFIS and ANFIS-DE in 20
forecasting monthly evapotranspiration rates under heterogeneous 21
climatic conditions 22
23
Abstract 24
Reference crop evapotranspiration (ET0) is one of the most important hydro-climatological 25
components which directly affects agricultural productions, and its forecasting is critical for water 26
managers and irrigation planners. In this study, adaptive neuro-fuzzy inference system (ANFIS) 27
model has been hybridized by differential evolution (DE) optimization algorithm as a novel 28
approach to forecast monthly ET0. Furthermore, this model has been compared with the classic 29
stochastic time series model. For this, the ET0 rates were calculated on monthly scale during 1995-30
2018, based on FAO-56 Penman-Monteith equation and meteorological data including: minimum 31
air temperature, maximum air temperature, mean air temperature, minimum relative humidity, 32
maximum relative humidity & sunshine duration. The investigation was performed on 6 stations 33
in different climates of Iran, including: Bandar Anzali & Ramsar (per-humid), Gharakhil (sub-34
humid), Shiraz (semi-arid), Ahwaz (arid) and Yazd (extra-arid). The models’ performances were 35
evaluated by the criteria percent bias (PB), root mean squared error (RMSE), normalized RMSE 36
(NRMSE) and Nash-Sutcliff (NS) coefficient. Surveys confirm the high capability of the hybrid 37
ANFIS-DE model in monthly ET0 forecasting; so that the DE algorithm was able to improve the 38
accuracy of ANFIS, by 16% on average. Seasonal autoregressive integrated moving average 39
(SARIMA) was the most suitable pattern among the time series stochastic models, and superior 40
compared to its other competitors. Consequently, due to the simplicity and parsimony, the 41
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SARIMA was suggested more appropriate for monthly ET0 forecasting in all the climates. 42
Comparison between the different climates confirmed that the climate type significantly affects 43
the forecasting accuracies: it’s revealed that all the models work better in extra-arid, arid and semi-44
arid climates, than the humid and per-humid areas. 45
Keywords: Differential Evolution; ANFIS; Stochastic; ARIMA; Time series prediction; 46
Reference Crop Evapotranspiration 47
1. Introduction 48
The process of water parting the surface of moist soil is called evaporation, whereas this 49
phenomenon from leaves’ pores is called transpiration. Since recognizing these two phenomena 50
on farms is not easy, they are to be considered as one integrated single variable referred to as 51
"evapotranspiration”. On the other hand, evapotranspiration is considered as the water requirement 52
of plants, so its measurement is very important in all agricultural and irrigation projects. The 53
amount of evapotranspiration is measured by a lysimeter. Due to the sensitivity of the lysimeter, 54
there is a need for the presence of a technician expert on-site in order for the lysimeter to be 55
continuously calibrated. Consequently, if good care is not taken, the recorded cases of lysimeter 56
may have errors. As a remedy, the International Commission on Irrigation and Drainage (ICID) 57
and World Meteorological Organization (WMO) have recognized the FAO-56 Penman-Monteith 58
equation (FAO-56 PM), as a suitable alternative to the lysimeter (Allen et al., 1998); which can 59
use several meteorological variables to estimate the evapotranspiration rate with an acceptable 60
accuracy. 61
In recent years, despite the presence of some well-known mathematical models such as Penman-62
Monteith, Thornthwaite, Hargreaves-Samani, Blaney-Criddle, etc., the black-box artificial 63
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intelligence (AI) models have been able to show acceptable accuracy in estimating 64
evapotranspiration. For example, Mohammadi & Mehdizadeh (2020) and Ahmadi et al. (2021) by 65
carrying out a survey on the arid and semi-arid regions of Iran found that in the complete absence 66
of meteorological variables (which are required to use the Penman method), the AI models are 67
able to estimate evapotranspiration with reasonable accuracy, by the least available meteorological 68
variables. They also contended that integrating AI models with bio-inspired optimization 69
algorithms can significantly increase the accuracy of evapotranspiration estimation. In Australia, 70
AIs were able to provide an accurate estimate of evapotranspiration with only temperature and 71
wind speed as available variables (Falamarzi et al., 2014); which in the absence of complete 72
meteorological variables can be considered as suitable alternative for the FAO-56 PM model. Also, 73
in cases such as Kumar et al. (2002), the validation of the estimated evapotranspiration from neural 74
networks using lysimeter measured evapotranspiration values, and comparing them with the 75
outputs of the FAO-56 PM model showed that AIs can be a better estimator for evapotranspiration. 76
Reference crop evapotranspiration (ET0) is one of the main components of the hydrological cycle 77
associated with agricultural systems. Accurate estimation and prediction of ET0 is very important 78
in water resources management, irrigation planning, and determining the water needs of plants. 79
Forecasting the evapotranspiration rates, through providing information on the future status of 80
evapotranspiration at different time scales can be of great help in making appropriate decisions, 81
planning as well as applying management methods of water resources. Data-driven models such 82
as stochastic and artificial intelligence methods are efficient approaches that have shown good 83
performance in modeling and predicting hydrometeorological variables in recent years (Aghelpour 84
et al., 2021c; Mohammadi et al., 2020; Aghelpour et al. 2020b). Karbasi (2018) used AIs in 85
forecasting ET0 for 1, 2, 3, 7, 10, 14, 18, 24, and 30 days’ horizons. Karbasi (2018) concluded that 86
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the predictions’ accuracy was desirable and showed that with increasing the forecast horizon, the 87
forecasting accuracy decreases. A comparison between stochastic and artificial intelligence 88
methods in Spain revealed that both model types predicted weekly evapotranspiration effectively 89
(Landeras et al., 2009). Lucas et al. (2020) compared the Seasonal Autoregressive Integrated 90
Moving Average (SARIMA) stochastic model with the Convolutional Neural Network (CNN) 91
model in order to predict daily evapotranspiration in Brazil. They concluded that the CNN model 92
is able to provide a more accurate prediction of evapotranspiration than the SARIMA model. In 93
opposite, in the Tamil Nadu of India, a comparison was made between artificial intelligence and 94
stochastic methods and stochastic models were introduced more appropriate for predicting ET0 95
(Kishore & Pushpalatha, 2017). Predicting evapotranspiration especially in areas such as Iran 96
which facing limited water resources, is doubly important for the determination of the cultivation 97
pattern, and proper management of water and soil resources. In Iran, these two types of numerical 98
models (stochastics and AIs) have been used to predict ET0. Ashrafzadeh et al. (2020) used the 99
SARIMA, Group Method of Data Handling (GMDH), and Support Vector Machine (SVM) 100
models, to predict ET0 in humid areas of the Caspian Sea’s southern margin. They evaluated the 101
accuracy of the models and showed that the mentioned models are able to predict the ET0 value 102
for the next 2 years, with the same suitable accuracy as the train-test period. 103
The Adaptive Neuro-Fuzzy Inference System (ANFIS) model is one of the most efficient AI 104
methods that has been used in both simple and hybridized forms, for hydrological and 105
meteorological modeling. ANFIS model showed its acceptable performances, in solar radiation 106
estimation (Üstün et al., 2020; Benmouiza & Cheknane, 2019; Halabi et al., 2018; Khosravi et al., 107
2018), pan evaporation estimation (Adnan et al., 2019; Guven & Kisi, 2013; Keskin et al., 2009), 108
drought forecasting (Aghelpour et al., 2021a; Aghelpour et al., 2021b; Aghelpour et al., 2020a; 109
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Aghelpour et al., 2020c; Kisi et al., 2019), river flow forecasting (Aghelpour & Varshavian, 2020; 110
Allawi et al., 2018), rainfall forecasting (Mekanik et al., 2016; Yaseen et al., 2018; Aghelpour et 111
al., 2021d) and wind speed forecasting (Maroufpoor et al., 2019). However, they are rarely used 112
in evapotranspiration prediction studies. The combination of bio-inspired optimization algorithms 113
has improved the performances of AIs in most cases, significantly (Deo et al., 2018; Aghelpour et 114
al., 2019; Paham et al., 2021; Aghelpour & Varshavian, 2021; Mohammadi et al., 2021). These 115
algorithms that use complex evolutionary methods can optimally enhance the parameters of AIs, 116
and significantly increase the accuracy of estimates and predictions (Moazenzadeh & 117
Mohammadi, 2019; Ashrafzadeh et al., 2019; Ashrafzadeh et al. al., 2020; Aghelpour et al., 118
2020c). 119
The present study intends to use the ANFIS model to predict the reference evapotranspiration and 120
compare it with the classical SARIMA stochastic model. Moreover, as a novelty, the Differential 121
Evolution (DE) algorithm (a bio-inspired algorithm) which is hybridized with the ANFIS model, 122
has been used as ANFIS-DE to optimize and improve the ANFIS’s prediction accuracy. In this 123
study, stations from different climates (from extra-arid to per-humid) are studied and for the first 124
time, the effect of climate type is also investigated on the accuracy of the models predicting ET0; 125
which is another novelty aspect of the current research. 126
2. Materials and methods 127
2.1.Data and areas under investigation 128
Iran is located in the Middle East, on the dry belt of the earth. Consequently, it is facing limited 129
water resources in human life’s different sectors, such as agriculture. According to De-Martonne 130
climatic zoning, Iran has 28 different climatic classes (Rahimi et al., 2013; Aghelpour et al., 131
2020a). The majority of regions of Iran have arid (central desert, southwest, and southwest of the 132
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country) and semi-arid climates (The Zagros Mountains in the west and northwest of the country 133
as well as northeastern regions), and only small areas of Iran have humid climates (Southern shore 134
of the Caspian Sea in the north). The rate of evapotranspiration, which is affected by different 135
meteorological factors, varies in different climatic zones. For example, in arid regions such as 136
Ahwaz, the range of ET0 is between 40 and 350 mm per month, while in humid climates like 137
Ramsar, the ET0 varies between 20 and 158 mm per month. This paper aims to investigate the 138
effect of the type of the climate on the accuracy of models predicting evapotranspiration. For this, 139
six synoptic stations from different climates of Iran are considered (Figure 1). 140
<Figure 1. here> 141
Three stations were selected from humid and sub-humid areas of northern Iran (on the southern 142
margin of the Caspian Sea), and the other three stations were selected from arid and semi-arid 143
areas in central and southwestern parts of Iran. Most of the agricultural lands in the northern humid 144
areas are under rice cultivation and the horticultural lands in this area are often under citrus 145
cultivation. In arid and semi-arid regions of the southern parts of Iran, the main agricultural crops 146
include wheat and maize, and the important horticultural crops are grapes and pistachios. A 147
summary of information on the climatic zones in this study, stations, and common products in 148
them is shown in Table 1. 149
<Table 1. here> 150
The data used in this paper include monthly meteorological data and belong to the period 1995-151
2018. These data include minimum air temperature (Tmin), maximum air temperature (Tmax), 152
mean air temperature (Tmean), minimum relative humidity (RHmin), maximum relative humidity 153
(RHmax) and sunshine duration (SSD), which are prepared on a monthly scale of the Iranian 154
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Meteorological Organization (IRIMO). Using these data and FAO-56 PM model, the amount of 155
monthly evapotranspiration was estimated in the 6 mentioned stations. The “Evapotranspiration” 156
package in R software was used to estimate the evapotranspiration rates, based on the FAO-56 PM 157
method. For modeling, the period under study was divided into two parts of training and testing, 158
which include 75% (the first 18 years) and 25% (the remaining 6 years), respectively. The 159
characteristics of the meteorological data as well as the estimated evapotranspiration data are 160
shown in Table 2. 161
<Table 2. here> 162
2.2.Time series model 163
A time series is a set of recorded observations of a variable such as Xi overtime in the form of X1, 164 X2, X3, …, XN between which the time interval is equal (Gutam & Sinha, 2016). Time series 165
models are kind of stochastic models that work based on regression coefficients and use the time 166
lags of the target variable, as the model’s input variable. These models include Autoregressive 167
(AR), Integrated (I), and moving average (MA) components. They are shown in an integrated state 168
known as Autoregressive Integral Moving Average (ARIMA). The Seasonal ARIMA (SARIMA) 169
model is a model that can be used for numerical simulation of the stochastic behavior of periodic 170
time series. In other words, SARIMA is a linear parametric stochastic model which can be used to 171
model and predict variables, which have seasonal autocorrelations. The cross form of this model 172
is shown as SARIMA(p, d, q)×(P, D, Q)ω; in which ω is the periodicity; p, d, and q are the non-173
seasonal degrees of autoregressive, differencing and moving average, respectively; P, D, and Q 174
are the seasonal degrees of autoregressive, differencing and moving average, respectively. The 175
general form of this model is shown below: (Salas et al, 1980): 176
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Eq. 1 𝛷𝑃(𝐵𝜔)𝜙𝑝(𝐵)𝛻𝜔𝐷𝛻𝑑𝑋𝑡 = 𝜃𝑞(𝐵)𝛩𝑄(𝐵𝜔)𝜀𝑡 In this formula 𝑋𝑡 is a stochastic variable as the target and 𝜀𝑡 is a normal random variable with 177
mean μ and variance 𝜎𝜀2, as a residual. Parameters of B including Φ, ϕ, 𝛻𝜔𝐷, 𝛻𝑑, Θ, θ, represent the 178
backward operators associated with seasonal autoregressive, non-seasonal autoregressive, 179
seasonal differencing and non-seasonal differencing, seasonal moving average and non-seasonal 180
moving average, respectively. Whose equations are described in equations 2 to 7 (Salas et al, 181
1980). 182
Eq. 2 𝛷𝑃(𝐵𝜔) = (1 − 𝛷1𝐵𝜔×1 − ⋯− 𝛷𝑃𝐵𝜔×𝑃)
183
Eq. 3 𝜙𝑝(𝐵) = (1 − 𝜙1𝐵1 − ⋯− 𝜙𝑝𝐵𝑝)
184
Eq. 4 𝛻𝜔𝐷 = (1 − 𝐵𝜔)𝐷
185
Eq. 5 𝛻𝑑 = (1 − 𝐵)𝑑
186
Eq. 6 𝛩𝑄(𝐵𝜔) = (1 − 𝛩1𝐵𝜔×1 − ⋯− 𝛩𝑄𝐵𝜔×𝑄)
187
Eq. 7 𝜃𝑞(𝐵) = (1 − 𝜃1𝐵1 − ⋯− 𝜃𝑞𝐵𝑞)
In this research, the Minitab software and the SARIMA model have been used to simulate and 188
predict evapotranspiration time series. 189
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2.3.Adaptive Neuro-Fuzzy Inference System (ANFIS) 190
ANFIS model has the ability to make relationships between input and output data using fuzzy rules 191
and to learn from a neural network in order to generate input structure for a system. ANFIS model 192
designs and creates non-linear maps to define relationships between input and output spaces by 193
employing Artificial Neural Network (ANN) and fuzzy logic, which is known as a neuro-fuzzy 194
system. Fuzzy systems include three different parts, namely fuzzification, inference engine, and 195
defuzzification. Fuzzy rules are achieved by utilizing fuzzy inference systems. A Fuzzy inference 196
system consists of two different inferences, namely Mamdani and Sugeno. They both work in an 197
excellent fashion when they are combined with an optimization algorithm and adaptive techniques 198
(Khosravi et al., 2018). In this paper, we use Sugeno inference. Figure 2 shows the structure of the 199
ANFIS model. 200
<Figure 2. here> 201
These two equations are the base rules of Sugeno inference: 202
Eq. 8 Rule 1: 𝑖𝑓 𝑥 𝑖𝑠 𝐴1 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝐵1, 𝑡ℎ𝑒𝑛 𝑓1 = 𝑝1𝑥 + 𝑞1𝑦 + 𝑟1
203
Eq. 9 Rule 1: 𝑖𝑓 𝑥 𝑖𝑠 𝐴2 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝐵2, 𝑡ℎ𝑒𝑛 𝑓2 = 𝑝2𝑥 + 𝑞2𝑦 + 𝑟2
ANFIS model contains different layers. Layer one, in this model, is the fuzzification layer. Each 204
node receives a signal and then transfers it to the next layer. The following equation describes the 205
cells outputs (𝑂1𝑖 ) (Khosravi et al., 2018; Haznedar and Kalinli, 2016): 206
Eq. 10 𝑂1𝑖 = 𝜇𝐴𝑖(𝑥); 𝑖 = 1, 2 𝜇𝐴𝑖 is related to Membership Function (MF). 𝐴𝑖 is linguistic variable and it is related to node 207
function. The following equation shows the common formula for 𝜇𝐴𝑖 208
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Eq. 11 𝜇𝐴𝑖(𝑥) = 𝑒𝑥𝑝 {− [(𝑥 − 𝑐𝑖𝑎𝑖 )2]𝑏𝑖} In this equation, x is input and 𝑎𝑖, 𝑏𝑖, 𝑐𝑖 are premise parameters. Layer 2 is called the rule layer 209
which is obtained by membership degrees. All the output nodes establish the firing strength of a 210
fuzzy rule. 211
Eq. 12 𝑂2𝑖 = 𝑤𝑖 = 𝜇𝐴𝑖(𝑥) 𝜇𝐵𝑖(𝑦); 𝑖 = 1, 2
Layer 3 is the normalization layer. In this layer, all the nodes are fixed and they are tagged with 212
N. The rule's firing strength to the sum of all rules' firing strengths is the ratio that is calculated by 213
the 𝑖𝑡ℎ node in the normalization layer. 214
Eq. 13 𝑂3𝑖 = 𝑤𝑖̅̅ ̅ = 𝑤𝑖𝑤1 + 𝑤2 ; 𝑖 = 1, 2
The defuzzification layer is the layer 4 of ANFIS model. Each rule uses the value of the previous 215
layer to compute the output value. 216
Eq. 14 𝑂4𝑖 = 𝑤𝑖̅̅ ̅𝑓𝑖 = 𝑤𝑖̅̅ ̅(𝑝𝑖𝑥 + 𝑞𝑖𝑦 + 𝑟𝑖); 𝑖 = 1, 2
In this equation, 𝑤𝑖̅̅ ̅ comes from the previous layer, namely layer 3. 𝑤𝑖̅̅ ̅ is a normalized firing 217
strength and 𝑝𝑖, 𝑞𝑖, and 𝑟𝑖 are the consequent parameters. Layer 5 is called the sum layer. By 218
summing the output values of the rules that come from the previous layer, the final output of the 219
ANFIS model is calculated. 220
Eq. 15 𝑂5𝑖 = 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑜𝑢𝑡𝑝𝑢𝑡 = ∑𝑤𝑖̅̅ ̅𝑓𝑖𝑖 = ∑ 𝑤𝑖𝑓𝑖𝑖∑ 𝑤𝑖𝑖 𝑖 = 1, 2
To implement the ANFIS model, MATLAB software is used in this study. 221
To summarize, the ANFIS model contains two sets of parameters: premise parameters and 222
consequence parameters. Premise parameters are input parameters of MFs and their aim is to 223
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specify the shape and the location of the input MFs (parameters of input MFs). Consequence 224
parameters are output parameters of MFs (parameters of output MFs) (Jang, 1993). To estimate 225
these parameters, classical ANFIS uses the least square (LS) methods. However, in the current 226
research, we have developed a novel ANFIS-DE model, which uses the meta-heuristic Differential 227
Evolution (DE) algorithm to estimate ANFIS’s sets of parameters. 228
2.4. Differential Evolution (DE) optimization algorithm 229
Although Differential Evolution (DE) uses basic optimized operations such as mutation, crossover, 230
and selection, it is an impressive and powerful optimization algorithm. One of the privileges of 231
this algorithm is that it has parallel search methods and uses NP and also it has D-dimensional 232
vectors of parameters. The advantage of these vectors is that they do not change during the 233
minimization procedure. DE performs a population process for each generation G. First, one 234
population vector is randomly initialized including the parameters and this probability distribution 235
is uniformed. When preliminary solution is achieved, DE algorithm calculates the difference 236
between the weights of two population vectors and assigns it to the third vector in order to produce 237
new parameter vectors, which is known as the mutation operation (Halabi et al., 2018): 238
Eq. 16 𝑣𝑖,𝐺+1 = 𝑥𝑖,𝐺 + 𝐹(𝑥𝑟2,𝐺 − 𝑥𝑟3,𝐺)
According to 𝑣𝑖,𝐺+1, these mutant vectors, 𝑥𝑖, 𝐺 and 𝑖 = 1,2,3, … ,𝑁𝑃 are created, while 𝑟1, 𝑟2, 239
and 𝑟3 are randomly integers and NP is selected from this distribution: integers ∈ [1,2,3, … ,𝑁𝑃]. 240
Moreover, 𝐼 and 𝐹 are real values and they are different from each other ∈ [1,2,3, … ,𝑁𝑃]. 241
During the mixing process which is also called crossover operation, parameters of the mutated 242
vector are mixed with other vector parameters to create the trial vector. The following equations 243
describe this mixing process: 244
13
Eq. 17 𝑢𝑖,𝐺+1 = (𝑢1𝑖,𝐺+1, 𝑢2𝑖,𝐺+1, … , 𝑢𝑑𝑖,𝐺+1)
245
Eq. 18 𝑢𝑗𝑖,𝐺+1 = {𝑣𝑗𝑖,𝐺+1; 𝑖𝑓 𝑟𝑎𝑛𝑑𝑏(𝑗) ≤ 𝐶𝑅 𝑜𝑟 𝑗 = 𝑟𝑛𝑏𝑟(𝑖)𝑥𝑗𝑖,𝐺+1; 𝑖𝑓 𝑟𝑎𝑛𝑑𝑏(𝑗) > 𝐶𝑅 𝑜𝑟 𝑗 ≠ 𝑟𝑛𝑏𝑟(𝑖)
In this equation, 𝑢𝑖,𝐺+1 is the trailer and 𝑥𝑖,𝐺 is the target vector, where 𝑢𝑖,𝐺+1 and 𝑥𝑖,𝐺 are the 246
trailer and target vectors, respectively. 𝑟𝑎𝑛𝑑𝑏(𝑗) is the Jth uniform random evaluation ∈ [0.1], 247 𝑟𝑛𝑏𝑟(𝑖) is a random value index ∈ [1,2,3, … , 𝑑] and 𝐶𝑅 is crossover constant which is determined 248
by users. The selection operation is the last operation. The trial vector costs a lower cost function 249
than the target vector. Therefore, the selection operation uses the trial vector as a target value for 250
the next generation. 𝑁𝑃 competitions are considered like one generation procedure as each 251
population vector has to serve once as the target vector. Complementary descriptions about the DE 252
optimization algorithm can be found in Storn & Price (1997) and Halabi et al. (2018). The DE 253
algorithm flowchart is illustrated in Figure 3. 254
<Figure 3. here> 255
In this paper, the DE algorithm is implemented by coding in MATLAB software’s environment. 256
The trial and error method is used to choose the best operators of DE to optimize the ANFIS model. 257
They are illustrated in Table 3. 258
<Table 3. here> 259
2.5.Evaluating the accuracy of the predictions 260
This study uses six criteria to evaluate the performance of the models: Root Mean Square Error 261
(RMSE), Normalized RMSE (NRMSE), Percent Bias (PB), Pearson correlation coefficient (R), 262
coefficient of determination (R2), and Nash- Sutcliff coefficient (NS). In general, these criteria are 263
14
used to compare the accuracy of different models with each other. Furthermore, they are used to 264
compare the accuracy of models in different climates. To calculate them, we need two series of 265
predicted and observed evapotranspiration data. Their equations are as follows. 266
Eq. 19 𝑅𝑀𝑆𝐸 = √1𝑛 ∑(𝐸𝑇𝑂𝑖 − 𝐸𝑇𝑃𝑖)2𝑛𝑖=1 ; 0 < 𝑅𝑀𝑆𝐸 < +∞
267
Eq. 20 𝑃𝐵 = ∑(𝐸𝑇𝑂𝑖 − 𝐸𝑇𝑃𝑖𝐸𝑇𝑂𝑖 )𝑛𝑖=1 ; −∞ < 𝑃𝐵 < +∞
268
Eq. 21 𝑅 = ∑ (𝐸𝑇𝑂𝑖 − 𝐸𝑇𝑂̅̅ ̅̅ ̅̅ )𝑛𝑖=1 (𝐸𝑇𝑃𝑖 − 𝐸𝑇𝑃̅̅ ̅̅ ̅̅ )√∑ (𝐸𝑇𝑂𝑖 − 𝐸𝑇𝑂̅̅ ̅̅ ̅̅ )2𝑛𝑖=1 ∗ √∑ (𝐸𝑇𝑃𝑖 − 𝐸𝑇𝑃̅̅ ̅̅ ̅̅ )2𝑛𝑖=1 ; −1 < 𝑅 < 1
269
Eq. 22 𝑅2 = [ ∑ (𝐸𝑇𝑂𝑖 − 𝐸𝑇𝑂̅̅ ̅̅ ̅̅ )𝑛𝑖=1 (𝐸𝑇𝑃𝑖 − 𝐸𝑇𝑃̅̅ ̅̅ ̅̅ )√∑ (𝐸𝑇𝑂𝑖 − 𝐸𝑇𝑂̅̅ ̅̅ ̅̅ )2𝑛𝑖=1 ∗ √∑ (𝐸𝑇𝑃𝑖 − 𝐸𝑇𝑃̅̅ ̅̅ ̅̅ )2𝑛𝑖=1 ]
2 ; 0 < 𝑅2 < 1
270
Eq. 23 𝑁𝑅𝑀𝑆𝐸 = √1𝑛 ∑ (𝐸𝑇𝑂𝑖 − 𝐸𝑇𝑃𝑖)2𝑛𝑖=1𝐸𝑇𝑂𝑚𝑎𝑥 − 𝐸𝑇𝑂𝑚𝑖𝑛 ; 0 < 𝑁𝑅𝑀𝑆𝐸 < +∞
271
Eq. 24 𝑁𝑆 = 1 − ∑ (𝐸𝑇𝑂𝑖 − 𝐸𝑇𝑃𝑖)2𝑛𝑖=1∑ (𝐸𝑇𝑂𝑖 − 𝐸𝑇𝑂̅̅ ̅̅ ̅̅ )2𝑛𝑖=1 ; −∞ < 𝑁𝑆 < 1
15
𝐸𝑇𝑂𝑖 shows the amount of observed evapotranspiration of the ith month, 𝐸𝑇𝑃𝑖 is the amount of 272
evapotranspiration predicted in the ith month, 𝐸𝑇𝑂̅̅ ̅̅ ̅̅ shows the mean of observational 273
evapotranspiration, 𝐸𝑇𝑃̅̅ ̅̅ ̅̅ represents the average of the predictive evapotranspiration ، 𝐸𝑇𝑂𝑚𝑎𝑥 is 274
the maximum of the observational evapotranspiration, and finally 𝐸𝑇𝑂𝑚𝑖𝑛 is the minimum of the 275
observational evapotranspiration. According to the defined range for these criteria, the closer the 276
RMSE, PB and NRMSE are to zero, and the closer NS, R, and R2 are to one, the better the model 277
performance is. Another point about NRMSE is that it has 4 intervals in terms of evaluating the 278
quality of models: 1) NRMSE> 0.3 poor performance, 2) 0.2 <NRMSE <0.3 average performance 279
3) 0.1 <NRMSE <0.2 good performance and 4) 0 <NRMSE <0.1 excellent performance. 280
The general process of modeling and predicting the evapotranspiration time series in this paper is 281
shown as a flowchart in Figure 4. 282
<Figure 4 here> 283
3. Results 284
3.1.Modeling and evaluating the predictions 285
In this study, ET0’s monthly time lags were considered as input to the models. Therefore, 286
Autocorrelation Function (ACF) diagrams for different stations were considered (Figure 5), which 287
show the extent and significance of the correlation of the variable with its previous steps’ amounts. 288
<Figure 5. here> 289
As can be seen from Figure 5, the ET0 data in all 6 stations have a significant seasonal trend. The 290
ET0 time series are periodic and have a 12 months’ periodicity. To moderate this seasonal trend, 291
several degrees of seasonal differentiation with a lag of 12 months (equal to the periodicity) were 292
considered. Investigations showed that seasonal differentiation of order "one" has the best 293
16
consistency with ET0 data. As a result, the SARIMA model is modified as the SARIMA pattern 294
SARIMA (p, 0, q)(P, 1, Q)12. Moreover, when the time lag increases, the significance threshold of 295
correlation (dashed line) increases and more than three return periods (36 months), it reaches a 296
point that is practically logical not to use them as inputs. Therefore, a maximum lag of 36 months 297
is considered as inputs for all models. In the SARIMA model, this includes seasonal autoregressive 298
and moving average degrees (P & Q), which is equal to 1, 2, and 3. These degrees and also the 299
non-seasonal degrees of autoregressive and moving average (p & q) were all tested, and their best 300
performance was selected for each station and reported in Table 4. Simple and hybrid ANFIS 301
models (ANFIS & ANFIS-DE) were implemented based on the fuzzy c-means (FCM) clustering 302
method. Lags of 1, 6, 12, 18, 24, 30, and 36 months were also considered as inputs to these AI 303
models. 304
<Table 4. here> 305
In Table 4, the predictions of all three models were evaluated by the mentioned evaluation metrics. 306
Since the test section actually shows the validity of the models, the test section is also discussed 307
in the interpretations of this section. At first, it can be seen that in all stations, the R coefficients 308
are very high, which indicates the optimal performance of the models in predicting monthly ET0 309
(the minimum value of R is equal to 0.949, which belongs to the simple ANFIS model in Ramsar 310
station). Additionally, the amount of PB in all cases is very small (close to zero); which confirms 311
the lack of significant under/overestimation and consequently the excellent performance of the 312
models. According to Table 4, in all stations, the SARIMA linear model has superior performance 313
than the other two models, and the weakest performance among the models belongs to the simple 314
ANFIS model. The DE algorithm in combination with the ANFIS model (ANFIS-DE), was able 315
to increase the prediction accuracy for ANFIS by an average of 15.8%. The lowest prediction error 316
17
belongs to the SARIMA model at Shiraz station with RMSE = 7.918 𝑚𝑚𝑚𝑜𝑛𝑡ℎ. The highest prediction 317
error is reported in Ahwaz station with RMSE = 16.906𝑚𝑚𝑚𝑜𝑛𝑡ℎ , which belongs to the simple ANFIS 318
model. 319
3.2.Comparison between the models 320
Scatter plots are used for graphical illustration of the correlation between the predicted and actual 321
values of monthly ET0 (figure 6). 322
<Figure 6. here> 323
In Figure 6, the horizontal axis of the graphs represents the observed ET0 data, and the vertical 324
axis represents the predictions presented by the models. This figure shows that at all stations, the 325
slope of the fitted regression line between the observed-predicted data samples is very small 326
associated with the X = Y line. The points are well concentrated around their regression line, and 327
this concentration is more on the diagrams related to the SARIMA model than the other two 328
models. On the other hand, the R2 coefficient shows that the SARIMA linear model offers a better 329
prediction than the other two nonlinear and complex models, ANFIS and ANFIS-DE. Also, 330
ANFIS-DE predictions show better correlations compared to simple ANFIS. The diagrams in 331
Figure 6 show that the weakest performance belongs to the predictions of ANFIS in Ramsar (R2 = 332
0.901), and the best performance belongs to the predictions of SARIMA at Yazd station (R2= 333
0.984). In order to compare the models, the Taylor diagram is also represented for each station 334
(Figure 7). 335
<Figure 7. here> 336
18
This diagram (Figure 7) is able to simultaneously check the correlation, the error, and also to 337
compare their standard deviations, between the outputs of several models vs their observational 338
values. In these diagrams, point O is an indicator of observational data, and points A, B, and C are 339
the indicators of the SARIMA, ANFIS, and ANFIS-DE models, respectively. At all stations, point 340
A is located the closest to point O, confirming the superiority of the SARIMA model. After that, 341
ANFIS-DE (point C) and ANFIS (point B) models are in the second and third places, respectively. 342
The best position of points A, B, and C belongs to Shiraz station, where these points are located 343
between two circles RMSE = 5 𝑚𝑚𝑚𝑜𝑛𝑡ℎ and RMSE = 10
𝑚𝑚𝑚𝑜𝑛𝑡ℎ, and around the radius R = 0.99. At 344
Yazd station, a similar situation to Shiraz is observed. The weakest points’ position can belong to 345
Bandar Anzali station; where points A, B and C are farthest from point O, between circles of 346
RMSE = 10 𝑚𝑚𝑚𝑜𝑛𝑡ℎ and RMSE = 15
𝑚𝑚𝑚𝑜𝑛𝑡ℎ, and between two radii of R = 0.99 and R = 0.95. 347
Furthermore, comparing the standard deviations between outputs and the observations, reveals that 348
the points of the models, especially point A, are in a very good position relative to the quadrant 349
close to point O. This shows that the models, especially SARIMA, have been able to show good 350
ability in estimating the standard deviation of actual ET0 values. 351
3.3.Comparison of ET0 prediction accuracy among different climates 352
In general, the comparison between the stations in Figure 7 represents that the humid stations are 353
located in weaker ranges of error and correlation, than the arid stations. Also, according to Figure 354
6, in humid and sub-humid climates, the R2 value resulted from the SARIMA model is in the range 355
of 0.95 - 0.96, while in arid and semi-arid regions, it is in the range of 0.97 - 0.98. Therefore, it is 356
evident that ET0 is slightly better predicted in arid areas. However, due to the different range of 357
19
ET0 data in different climates (Table 2), it is better to consider the normalized RMSE (NRMSE) 358
criterion at stations for evaluation (Figure 8). 359
<Figure 8. here> 360
In Figure 8, the NRMSE and NS criteria for the test period were plotted together as a combo-361
graph. This diagram is drawn separately for all models at all stations. At first, it can be seen that 362
all models have a NS value greater than 0.9, which confirms the very good prediction of ET0 by 363
the models. Moreover, the NRMSE value in all stations is less than 0.1. According to the quality 364
classes defined for NRMSE (Aghelpour & Varshavian, 2020), the predictions for all climates in 365
this study are considered very reasonable. The visible trend of NS and NRMSE is similar across 366
stations. Both criteria indicate a better prediction of ET0 in arid and semi-arid climates. In other 367
words, if the NS level is increased at a station, the NRMSE level will decrease at the same station 368
(which is well illustrated in the combo-graph). Therefore, it can be said that both criteria achieved 369
similar results in comparing the accuracy of ET0 prediction among the climates. For example, in 370
the ANFIS-DE model for humid and sub-humid stations, the NRMSE is between 0.07 - 0.09 and 371
the NS is between 0.93 - 0.95, while for arid and semi-arid stations, NRMSE is between 0.04 - 372
0.06 and NS is between 0.97 - 0.98. In the combo-graph belonging to the SARIMA model, the 373
NRMSE value for humid and sub-humid areas is between 0.06 - 0.08 and the NS value is between 374
0.94 - 0.96, while for arid and semi-arid areas, the NRMSE is between 0.04 - 0.05 and the NS is 375
between 0.98 - 0.99. The comparison of the models is similar to the previous diagrams and tables; 376
which reported the SARIMA model more appropriate. The predictions provided by the models can 377
also be graphically seen in time-series plots (Figure 9), to see the overlaps. 378
<Figure 9. here> 379
20
4. Discussion 380
Research on the use of AIs to estimate and predict the reference evapotranspiration, as in this 381
paper, have evaluated the results of these models as favorable (Ahmadi et al., 2021; Ashrafzadeh 382
et al., 2020; Adamala et al., 2018; Abrishami et al., 2019). Also, the desirability of the accuracy of 383
time series models in the current study is similar to the research of Gautam & Sinha (2016), 384
Landeras et al. (2009), Psilovikos & Elhag (2013), Mossad & Alazba (2016), and Bouznad et al. 385
(2020), that have been conducted in different climatic regions. The superiority of time series 386
models over AIs in ET0 forecasting in Iran, has also been reported in Ashrafzadeh et al. (2020); 387
however, their study only addressed the humid northern climate. Additionally, Ashrafzadeh et al. 388
(2020) used non-hybridized models of artificial intelligence; while the current research showed 389
that the novel hybrid ANFIS-DE model can significantly increase the accuracy of the simple 390
ANFIS model. In Brazil, however, AIs provided a relatively more accurate prediction of ET0 than 391
time series models did (Lucas et al., 2020), which contradicts the results of the current study. The 392
reason for this contradiction could be due to the differences between the climatic conditions of the 393
studies’ regions. 394
In comparison, between the climates of the present study, the geographical location as well as the 395
physical systems involved can be factors influencing the accuracy of ET0 prediction. For example, 396
the humid regions of northern Iran are affected by Caspian atmospheric systems and various 397
western systems such as the Black Sea and the Mediterranean Sea; while the western and 398
southwestern regions of Iran (such as Shiraz, and Ahwaz) are only weakly affected by the two 399
systems of Saudi Arabia’s high-pressure and Sudan’s low-pressure. Susceptibility to a large 400
number of systems can disrupt the order of time series, reduce autocorrelation and consequently 401
lead to a poor prediction. This difference in the order of the ET0 series in different climates is 402
21
depicted in the diagrams of Figure 9. On the other hand, these three stations of Shiraz, Ahwaz and 403
Yazd, are located near the Subtropical High-Pressure Belt (SHPB) (latitude 30 degrees), which 404
can stabilize the weather regime in these areas and thus make the ET0 series more regular. By 405
moving away from the SHPB and approaching the latitudes of the northern humid regions, the 406
effects of the irregularity of the annual regime become more obvious and can eventually lead to a 407
relative increase in the prediction errors in these areas. 408
5. Conclusion 409
Studies have been shown that the water requirement of plants can be predicted with very good 410
accuracy by using the time lags of the evapotranspiration variable. The currently used data-driven 411
approaches could provide acceptable predictions of ET0, regardless of the various atmospheric 412
and physical factors that affect it. This result is similar in all currently studied climates. Despite 413
the significant improvement (about 16%) of the ANFIS model in combination with the Differential 414
Evolution optimization algorithm, it still fails to compete with the SARIMA linear model. The 415
reason may be as Ashrafzadeh et al. (2020) has reported, the linear autocorrelation is stronger than 416
nonlinear autocorrelation in the ET0 time series. Finally, the present study proposes time series 417
models to better predict ET0 for two reasons: 1) higher accuracy 2) the simplicity of use. Another 418
important conclusion of this paper is that the type of climate in a region significantly affects the 419
accuracy of predictor models of ET0: In the arid and semi-arid climates of southern Iran, ET0 was 420
predicted more accurately than the humid and sub-humid regions of northern Iran. Due to the high 421
accuracy and promising results of the present study, the use of these data-driven models to predict 422
the water needs of plants in other geographical areas is recommended. Moreover, the use of the 423
current models especially SARIMA and the hybrid ANFIS-DE has research value for long-term 424
and multi-ahead years prediction of monthly ET0. The use and comparison of stochastic, artificial 425
22
intelligence, and metaheuristic models in predicting ET0 on a daily scale can be an interesting 426
topic for study, suggested to future researchers in this field. 427
Funding Statement 428
This work was supported by the Bu-Ali Sina University Deputy of Research (Grant numbers 99-429
277). 430
Author's Contribution 431
Conceptualization, Pouya Aghelpour; methodology, Pouya Aghelpour, Vahid Varshavian; 432
software, Pouya Aghelpour, validation, Pouya Aghelpour, Vahid Varshavian, and Zahra Hamedi; 433
investigation, Pouya Aghelpour, and Vahid Varshavian; resources, Zahra Hamedi; data curation, 434
Pouya Aghelpour; writing—original draft preparation, Vahid Varshavian, Pouya Aghelpour, and 435
Zahra Hamedi; writing—review and editing, Vahid Varshavian, Pouya Aghelpour; visualization, 436
Zahra Hamedi; supervision, Vahid Varshavian. All authors have read and agreed to the published 437
version of the manuscript. 438
Ethics approval 439
Not applicable, because this article does not contain any studies with human or animal subjects. 440
Consent for publication 441
The Authors hereby consents to publication of the Work in any and all Springer publications 442
Data & Code Availability 443
The data & Code used to support the findings of this study are available from the first and 444
corresponding author upon request. 445
Conflicts of Interest 446
The authors declare that they have no conflicts of interest. 447
448
23
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38. Mekanik, F., Imteaz, M. A., & Talei, A. (2016). Seasonal rainfall forecasting by adaptive network-based fuzzy inference system (ANFIS) using large scale climate signals. Climate dynamics, 46(9-10), 3097-3111. https://doi.org/10.1007/s00382-015-2755-2
39. Moazenzadeh, R., & Mohammadi, B. (2019). Assessment of bio-inspired metaheuristic optimisation algorithms for estimating soil temperature. Geoderma, 353, 152-171. https://doi.org/10.1016/j.geoderma.2019.06.028
40. Mohammadi, B., Guan, Y., Aghelpour, P., Emamgholizadeh, S., Pillco Zolá, R., & Zhang, D. (2020). Simulation of Titicaca Lake Water Level Fluctuations Using Hybrid Machine Learning Technique Integrated with Grey Wolf Optimizer Algorithm. Water, 12(11), 3015. https://doi.org/10.3390/w12113015
41. Mohammadi, B., Guan, Y., Moazenzadeh, R., & Safari, M. J. S. (2021). Implementation of hybrid particle swarm optimization-differential evolution algorithms coupled with multi-layer perceptron for suspended sediment load estimation. Catena, 198, 105024. https://doi.org/10.1016/j.catena.2020.105024
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43. Mossad, A., & Alazba, A. A. (2016). Simulation of temporal variation for reference evapotranspiration under arid climate. Arabian Journal of Geosciences, 9(5), 386. https://doi.org/10.1007/s12517-016-2482-y
44. Pham, Q. B., Sammen, S. S., Abba, S. I., Mohammadi, B., Shahid, S., & Abdulkadir, R. A. (2021). A new hybrid model based on relevance vector machine with flower pollination algorithm for phycocyanin pigment concentration estimation. Environmental Science and Pollution Research, 1-16. https://doi.org/10.1007/s11356-021-12792-2
45. Psilovikos, A., & Elhag, M. (2013). Forecasting of remotely sensed daily evapotranspiration data over Nile Delta region, Egypt. Water resources management, 27(12), 4115-4130. https://doi.org/10.1007/s11269-013-0368-2
46. Rahimi, J., Ebrahimpour, M., & Khalili, A. (2013). Spatial changes of extended De Martonne climatic zones affected by climate change in Iran. Theoretical and applied climatology, 112(3), 409-418. https://doi.org/10.1007/s00704-012-0741-8
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48. Storn, R., & Price, K. (1997). Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization, 11(4), 341-359. https://doi.org/10.1023/A:1008202821328
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Table 1. The studied stations’ location, climate (according to extended De-Martonne classification [Rahimi et al., 2013]) and the main agricultural/horticultural products of their regions
Province Station
Coordinates Climate (based on
extended De-Martonne method)
Main products
Latitude - northern (degree)
Longitude - eastern
(degree)
Elevation (m) Agricultural Horticultural
Gilan Bandar Anzali
37.47 49.47 -26.2 Per humid(B)
- Moderate
rice cultivars; tobacco;
watermelon
tea; olive; citrus; kiwi; plum
Mazandaran Ramsar 36.90 50.67 -20.0
Per humid(A) - Moderate
rice cultivars; wheat; soy;
rapeseed
citrus; kiwi; ornamental flower;
plants Gharakhil 36.45 52.77 14.7
Sub-humid - Moderate
Khuzestan Ahwaz 31.33 48.67 22.5 Arid - Warm wheat; barley;
maize; legumes; rapeseed
vegetable; cucurbits; potato;
onion
Fars Shiraz 29.53 52.60 1484.0 Semi arid - Moderate
wheat; barley; sugar beet; maize
almonds, grapes, pomegranates,
damask rose; figs
Yazd Yazd 31.90 54.28 1237.2 Extra arid -
Cold
sorghum, fodder maize, millet,
legumes, alfalfa
pistachios, pomegranates,
apricots, saffron
Table 2. Specifications of the meteorological data used and the calculated ET0 on monthly scale
Station Variable Training period Testing Period
Min.* Max. Average STD. Min. Max. Average STD.
Bandar Anzali Tmin (°C) 0.80 25.40 14.41 6.85 3.10 26.10 14.80 6.84 Tmax (°C) 5.30 31.80 19.24 7.14 8.40 32.80 20.12 7.57 Tmean (°C) 3.00 28.40 16.82 6.99 5.80 29.30 17.46 7.18 RHmax (%) 81.20 96.90 92.21 3.09 81.50 96.50 91.68 3.73 RHmin (%) 54.80 84.10 73.11 5.72 53.90 84.40 71.76 7.04
SSD (ℎ𝑟𝑚𝑜𝑛𝑡ℎ) 28.50 337.60 161.74 73.68 40.40 339.70 163.78 82.92
ET0 (mm𝑚𝑜𝑛𝑡ℎ) 20.60 174.30 74.39 43.57 22.70 170.30 80.42 49.65
Ramsar Tmin (°C) 0.90 24.90 13.77 6.82 2.90 25.40 14.34 6.85 Tmax (°C) 7.10 31.50 19.93 6.86 9.20 32.50 20.43 7.23 Tmean (°C) 4.00 28.20 16.86 6.82 6.10 28.90 17.39 7.03 RHmax (%) 80.60 97.30 89.85 3.33 80.30 95.10 90.18 3.80 RHmin (%) 56.50 84.20 69.07 4.83 56.70 82.70 69.61 5.82
SSD (ℎ𝑟𝑚𝑜𝑛𝑡ℎ) 39.00 289.20 139.53 51.16 52.80 309.70 140.29 58.79
ET0 (mm𝑚𝑜𝑛𝑡ℎ) 20.90 158.50 71.52 37.90 23.20 151.70 72.77 42.10
Gharakhil Tmin (°C) -1.30 23.80 12.76 7.14 1.50 24.20 13.03 7.20 Tmax (°C) 8.10 34.80 21.98 7.14 11.70 34.70 22.58 7.35 Tmean (°C) 3.40 28.80 17.37 7.11 6.60 29.20 17.80 7.26 RHmax (%) 89.40 98.90 95.40 2.04 89.20 97.00 94.16 2.07 RHmin (%) 46.50 76.90 62.45 5.59 47.60 73.50 62.27 5.41
SSD (ℎ𝑟𝑚𝑜𝑛𝑡ℎ) 40.30 310.20 170.11 49.43 73.30 317.60 169.54 53.09
ET0 (mm𝑚𝑜𝑛𝑡ℎ) 23.40 164.40 78.10 40.16 20.20 169.70 80.22 44.70
Ahwaz Tmin (°C) 6.20 31.50 19.44 7.86 7.40 31.40 19.79 8.02 Tmax (°C) 14.70 48.10 33.60 10.59 17.40 48.90 34.15 10.24 Tmean (°C) 10.40 39.80 26.52 9.20 13.40 39.90 26.98 9.10 RHmax (%) 28.10 95.80 60.09 19.00 27.80 96.30 62.35 18.27 RHmin (%) 6.80 67.10 23.85 14.67 7.80 64.70 25.46 13.46
SSD (ℎ𝑟𝑚𝑜𝑛𝑡ℎ) 162.40 383.60 273.79 58.02 163.60 370.30 272.99 58.36
ET0 (mm𝑚𝑜𝑛𝑡ℎ) 40.20 354.50 169.06 93.21 44.80 310.50 161.89 85.55
Shiraz Tmin (°C) -2.00 24.20 10.95 7.46 -1.10 22.30 10.46 7.29 Tmax (°C) 9.40 40.10 26.33 9.17 11.70 40.10 26.90 8.85 Tmean (°C) 4.80 32.10 18.64 8.26 5.60 31.10 18.68 8.04 RHmax (%) 30.00 91.90 58.33 17.96 27.80 90.90 58.51 18.24 RHmin (%) 6.60 54.50 20.86 11.01 4.30 49.50 17.51 10.04
SSD (ℎ𝑟𝑚𝑜𝑛𝑡ℎ) 208.50 372.30 296.88 40.68 222.70 370.30 294.97 40.10
ET0 (mm𝑚𝑜𝑛𝑡ℎ) 37.90 251.40 133.79 64.01 44.70 224.50 129.44 60.15
Yazd Tmin (°C) -4.40 28.30 13.24 8.74 1.10 27.40 14.32 8.46 Tmax (°C) 4.80 42.60 27.33 9.62 12.40 41.80 27.87 9.05 Tmean (°C) 0.20 35.50 20.29 9.16 6.80 34.60 21.10 8.74 RHmax (%) 15.50 87.70 41.06 19.22 12.60 80.40 38.11 17.38 RHmin (%) 5.10 57.60 16.25 9.96 4.90 39.60 14.49 7.54
SSD (ℎ𝑟𝑚𝑜𝑛𝑡ℎ) 209.80 376.80 292.77 47.08 200.40 383.00 296.97 47.65
ET0 (mm𝑚𝑜𝑛𝑡ℎ) 34.00 289.10 156.13 73.86 55.30 273.50 155.87 70.35
*Min. = Minimum; Max. = Maximum; STD = Standard deviation
Table 3. The operators of differential evolution Algorithm
Operator Value
Population 100 Maximum Number of Iterations 200
Crossover probability 0.1 Scaling factor lower bound 0.2 Scaling factor upper bound 0.8
Table 4. Evaluating the models’ predictions by evaluation criteria
Station Model
Train Test
RMSE
(𝑚𝑚𝑚𝑜𝑛𝑡ℎ) PB R
RMSE
(𝑚𝑚𝑚𝑜𝑛𝑡ℎ) PB R
Bandar Anzali SARIMA(1,0,0)(2,1,2)12
* 9.436 -0.026 0.977 10.078 -0.042 0.982 ANFIS 8.177 -0.014 0.983 12.767 0.035 0.970 ANFIS-DE 10.492 -0.019 0.971 10.532 -0.018 0.977
Ramsar SARIMA(1,0,2)(3,1,3)12 8.973 -0.011 0.973 9.711 -0.028 0.975 ANFIS 8.130 -0.011 0.977 13.257 -0.013 0.949 ANFIS-DE 11.171 -0.015 0.957 10.998 -0.013 0.965
Gharakhil SARIMA(1,0,0)(3,1,1)12 10.909 -0.013 0.963 9.713 -0.041 0.979
ANFIS 9.624 -0.014 0.971 12.569 -0.018 0.960 ANFIS-DE 12.300 -0.018 0.953 10.711 -0.005 0.970
Ahwaz SARIMA(1,0,1)(2,1,3)12 14.844 -0.003 0.987 12.789 0.020 0.990
ANFIS 12.597 -0.008 0.991 16.906 -0.021 0.983 ANFIS-DE 16.134 -0.008 0.984 14.533 -0.020 0.985
Shiraz SARIMA(1,0,1)(2,1,2)12 8.364 -0.004 0.991 7.918 0.013 0.992 ANFIS 6.281 -0.004 0.995 9.920 -0.007 0.986 ANFIS-DE 10.408 -0.009 0.987 9.077 -0.014 0.988
Yazd SARIMA(2,0,0)(3,1,3)12 10.142 -0.007 0.991 8.897 0.005 0.994 ANFIS 8.858 -0.008 0.993 10.537 0.007 0.989 ANFIS-DE 11.224 -0.011 0.989 9.548 0.000 0.991
*Bold rows specify the best fitted model in each station.
Figure 1. Location of the stations under investigation on the country
N
N
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x
x y
x y
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f
Figure 2. The schematic structure of an ANFIS model with two inputs
StartGenerate mutant vector for a new
population vectors
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EndYesNo
Figure 3. Flowchart of the optimization process based on differential evolution algorithm
Input phase
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ET0t
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ET0t-2
.
.
.
ET0t-n
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PB
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• ANFIS-DE’s predictions
Reporting the
most appropriate
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Differential Evolution (DE)
optimization
ANFIS
Modeling phase
Trees
GrassSoil
EvaporationTranspiration
Evapotranspiration of the next month (ET0t+1)
Conclusion Phase
Figure 4. General flowchart of the evapotranspiration modeling, prediction and evaluation processes
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Figure 6. Scatter plots to investigate the models’ predictions against their simultaneous observed values; the alphabets within the brackets refer to the stations: (a) Bandar Anzali, (b) Ramsar, (c) Gharakhil, (d)
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R² = 97.28% R² = 97.71%R² = 98.33%
R² = 97.88% R² = 98.19%R² = 98.41%
Figure 6. Continued
Bandar Anzali station
Ramsar station
Gharakhil station
Ahwaz station
Shiraz station
Yazd station
RMSE limits
(O)
Observational ET0
Legend
(A)
Predicted ET0 by SARIMA
(B)
Predicted ET0 by ANFIS
(C)
Predicted ET0 by ANFIS-DE
Figure 7. Taylor diagrams to compare the models in the stations; the diagram of each station is specified by its own name
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Figure 8. Combo-graph of NRMSE and NS criteria to make a comparison between the different climates
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Figure 9. Multiple tim
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odels’ predictions
